Chapter 6: Vector Analysis

As in Problem 17, find the following gradients in two ways and show that your answers are equivalent. $\mathbf{\nabla }\mathit{y}$

If A and B are unit vectors with an angle θ between them, and is a unit vector perpendicular to both A and B , evaluate $\left[\left(A\times B\right)\times \left(B\times C\right)\times \left(C\times A\right)\right]$

Find the gradient of $\mathit{w}\mathbf{=}{\mathit{x}}^{\mathbf{2}}{\mathit{y}}^{\mathbf{3}}\mathit{z}$ at $\left(1,2,-1\right)$.

Evaluate each of the following integrals in the easiest way you can.

$\mathbf{\oint }\left(2ydx-3xdy\right)$around the square bounded by $\mathbf{x}\mathbf{=}\mathbf{3}\mathbf{,}\mathbf{x}\mathbf{=}\mathbf{5}\mathbf{,}\mathbf{y}\mathbf{=}\mathbf{}\mathbf{1}\mathbf{}\mathbf{andy}\mathbf{=}\mathbf{3}$

If$\mathbf{A}\mathbf{=}\mathbf{2}\mathbf{i}\mathbf{-}\mathbf{j}\mathbf{-}\mathbf{k}$,role="math" localid="1659148191947" $\mathbf{B}\mathbf{=}\mathbf{2}\mathbf{i}\mathbf{-}\mathbf{3}\mathbf{j}\mathbf{+}\mathbf{k}\mathbf{,}$find$\left(A\times B\right)\mathbf{C}\mathbf{,}\mathbf{}\mathbf{,}\mathbf{A}\left(B\times C\right)\mathbf{,}\left(A\times B\right)\mathbf{\times }\mathbf{C}\mathbf{,}\mathbf{A}\mathbf{\times }\left(B\times C\right)\mathbf{,}\left(A\times B\right)\mathbf{\times }\mathbf{C}\mathbf{,}\mathbf{A}\mathbf{\times }\left(B\times C\right)\mathbf{.}$

For Problems 2 to 6, given$\mathbf{A}\mathbf{=}\mathbf{i}\mathbf{+}\mathbf{j}\mathbf{-}\mathbf{2}\mathbf{k}\mathbf{,}\mathbf{B}\mathbf{=}\mathbf{2}\mathbf{i}\mathbf{-}\mathbf{j}\mathbf{+}\mathbf{3}\mathbf{k}\mathbf{,}\mathbf{C}\mathbf{=}\mathbf{j}\mathbf{-}\mathbf{5}\mathbf{k}$

$$\begin{gathered} \mathbf{Evaluate}\mathbf{}\mathbf{the}\mathbf{}\mathbf{line}\mathbf{}\mathbf{integral}\mathbf{}\mathbf{\int }\left(x^2-y^2\right)\mathbf{dx}\mathbf{-}\mathbf{2}\mathbf{xydy}\mathbf{}\mathbf{along}\mathbf{}\mathbf{each}\mathbf{}\mathbf{of}\mathbf{}\mathbf{the}\mathbf{}\mathbf{following}\mathbf{}\mathbf{paths}\mathbf{}\mathbf{from}\mathbf{}\mathbf{(}\mathbf{0}\mathbf{,}\mathbf{}\mathbf{0}\mathbf{)}\mathbf{}\mathbf{to}\mathbf{}\mathbf{(}\mathbf{1}\mathbf{,}\mathbf{}\mathbf{2}\mathbf{)}\mathbf{.}\mathbf{} \\ \mathbf{\left(}\mathbf{a}\mathbf{\right)}\mathbf{}\mathbf{y}\mathbf{=}\mathbf{2}{\mathbf{x}}^{\mathbf{2}}\mathbf{}\mathbf{}\mathbf{.} \\ \mathbf{\left(}\mathbf{b}\mathbf{\right)}\mathbf{}\mathbf{x}\mathbf{=}{\mathbf{t}}^{\mathbf{2}} \\ \mathbf{\left(}\mathbf{c}\mathbf{\right)}\mathbf{}\mathbf{y}\mathbf{=}\mathbf{}\mathbf{0}\mathbf{}\mathbf{from}\mathbf{}\mathbf{x}\mathbf{}\mathbf{=}\mathbf{}\mathbf{0}\mathbf{}\mathbf{to}\mathbf{}\mathbf{x}\mathbf{}\mathbf{=}\mathbf{}\mathbf{2}\mathbf{;}\mathbf{}\mathbf{then}\mathbf{}\mathbf{along}\mathbf{}\mathbf{the}\mathbf{}\mathbf{straight}\mathbf{}\mathbf{line}\mathbf{}\mathbf{joining}\mathbf{}\mathbf{(}\mathbf{2}\mathbf{,}\mathbf{}\mathbf{0}\mathbf{)}\mathbf{}\mathbf{to}\mathbf{}\mathbf{(}\mathbf{1}\mathbf{,}\mathbf{}\mathbf{2}\mathbf{)}\mathbf{.} \end{gathered}$$

Do case (b) of Example $\mathbf{1}\mathbf{}$above.

Write out the equations corresponding to (9.3) and (9.4) for Q=2Xbetween points 3 and 4 in Figure 9.2, and add them to get (9.6).

$\iint \mathrm{P}\times \mathrm{nd\sigma }$over the upper half of the sphere $\mathrm{r}=1$, if $\mathrm{P}=\mathrm{curl}(\mathrm{jx}-\mathrm{kz})$.

For motion near the surface of the earth, we usually assume that the gravitational force on a mass m is

$\mathit{F}\mathbf{=}\mathbf{-}\mathit{m}\mathit{g}\mathit{k}$

but for motion involving an appreciable variation in distance r from the center of the earth, we must use

$\mathit{F}\mathbf{=}\mathbf{-}\frac{\mathbf{C}}{{\mathbf{r}}^{\mathbf{2}}}{\mathit{e}}_{\mathbf{r}}\mathbf{=}\frac{\mathbf{C}}{{\mathbf{r}}^{\mathbf{2}}}\frac{\mathbf{r}}{\left|r\right|}\mathbf{=}\mathbf{-}\frac{\mathbf{C}}{{\mathbf{r}}^{\mathbf{3}}}\mathit{r}$

where C is a constant. Show that both these F’s are conservative, and find the potential for each.